I
i
1913]
CERTAIN PROBLEMS RELATING TO THE POTENTIAL
159
The significance of the first three terms is brought out if we suppose that r is constant (a), so that the last term vanishes. In this case the exact solution is
a + p , a + a. - r-
,
log 0
whence
-. > M.
-1=^- +
T a
1.2
1.2.3
, (11)
.(12)
a \ a / in agreement with (10).
In the above investigation -^ is supposed to be zero exactly upon the circle of radius a. If the circle whose centre is taken as origin of coordinates be merely the circle of curvature of the curve fy = 0 at the point (8 = 0) under consideration, -ty- will not vanish exactly upon it, but only when r has the approximate value c03, c being a constant. In (6) an initial term R0 must be introduced, whose approximate value is —c9'ARi. But since -R0" vanishes with d, equation (7) and its consequences remain undisturbed and (10) is still available as a formula of interpolation. In all these cases, the success of the approximation depends of course upon the degree of slowness with which y, or r, varies.
Another form of the problem arises when what is given is not a pair of neighbouring curves along each of which (e.g.) the stream-function is constant, but one such curve together with the variation of potential along it. It is then required to construct a neighbouring stream-line and to determine the distribution of potential upon it, from which again a fresh departure may be made if desired. For this purpose we regard the rectangular coordinates x, y as functions of f (potential) and 77 (stream-function), so that
in which we are supposed to know f(£) corresponding to 77 = 0, i.e., so and y are there known functions of £. Take a point on 77 = 0, at which without loss of generality £ may be supposed also to vanish, and form the expressions for x and y in the neighbourhood. From
. K -f iy = A0 + iB0 + (A! + iBi) (£ + iy) + (Az we derive as — A0 + Al% — B-M + A% (£2 — ^
if)
= An
+ As%3 + Aamical similarity for viscous fluids were formulated in this memoir. Beynolda's important application was 80 years later.b)J1(k1b) ' 4&1F0'(M)